Throughout this year, I have had relatively good experiences with using In-Context-Learning [ICL] with AI chatbots and math in the fields of Linear Algebra, multivariate calculus and statistics, quadratic surfaces and respective examples. However, there are still situations where some AI-models fall into traps. I just want to name them for people using AI for math regularly.

Problem 1: Confusion of passive with active rotations and the choice of respective matrices during transformations

One of these situations occurs when dealing with passive and active rotations in combination with the right-hand-rule. E.g., both GPT Luna 6 and Mistral/Vibe have made mistakes when they used coordinate transformations during intensive reasoning for achieving solutions to admittedly complex tasks. E.g., it happened that the AI decided to chose a certain coordinate system to simplify a given task. But sometimes transformations between different coordinate systems are inherently required to solve a major problem. The AI will sooner or later detect this requirement ad make a choice regarding the transformation matrices which it thinks are the correct ones. Then there is always a chance that the AI may pick the wrong matrices. Which introduces at least sign errors – in the best case.

For tasks whose solution foreseeably require switches between rotated coordinate systems, I recommend not to let the bot solve the complex problem, which may require many successive steps, in one extensive reasoning process. Instead work step by step and force the bot to show and discuss the used rotation or other orthogonal transformation matrices explicitly before proceeding. The other option is to define a standard approach and respective conventions regarding matrices for your class of problems. This is what I have done with Mistral and GPT (as long as I have used it).

Problem 2: The tempting use of coordinates as parameters which correspond to non-orthogonal base vectors in another frame

Sometimes one has to deal with objects in a rotated lower-dimensional sub-space of a multidimensional space and its Cartesian coordinate system. This sub-space my contain the origin. You may need to describe internal vectors in the sub-space. Then you need a parameterization of the vectors within the sub-space and with respect to certain base vectors spanning it. (Think e.g. of a horizontal elliptic cylinder and an arbitrarily slicing plane through the origin in three dimensions.)

If both the sub-space and the encompassing coordinate frame use the same origin, it is tempting to use the original coordinates also for a parameterization of the subspace-vectors.( In the named example you might e.g. use x– and y-values to characterize vectors in the plane.) I have seen at least two cases where GPT Luna decided to do so. Such an approach is fully legitimate.

However, due to the arbitrary orientation of the sub-space this corresponds to the choice of non-orthogonal base vectors of the sub-space. Which has consequences for the metric and also eigenvalues of matrices. GPT forgot this fact and therefore also forgot that the so called Gram matrix has to be applied when solving the eigenvalue problem for certain matrices. Among other things this led to a wrong description of the dimensions of objects in the sub-space.

It is better in such cases not to leave the choice of base vectors for the description of in-subspace vectors to the AI. To avoid non-orthogonality pitfalls give the AI bot a hint or explicitly request the choice of orthogonal base vectors to span the sub-space.

Hope, this helps some folks using AI for simple undergraduate math and physics problems.